(a) Let $X$ and $Y$ be Banach spaces, and let $T: X \rightarrow Y$ be a surjective linear map. Assume that there is a constant $c>0$ such that $\|T x\| \geqslant c\|x\|$ for all $x \in X$. Show that $T$ is continuous. [You may use any standard result from general Banach space theory provided you clearly state it.] Give an example to show that the assumption that $X$ and $Y$ are complete is necessary.

(b) Let $C$ be a closed subset of a Banach space $X$ such that

(i) $x_{1}+x_{2} \in C$ for each $x_{1}, x_{2} \in C$;

(ii) $\lambda x \in C$ for each $x \in C$ and $\lambda>0$;

(iii) for each $x \in X$, there exist $x_{1}, x_{2} \in C$ such that $x=x_{1}-x_{2}$.

Prove that, for some $M>0$, the unit ball of $X$ is contained in the closure of the set

$$\left\{x_{1}-x_{2}: x_{i} \in C, \quad\left\|x_{i}\right\| \leqslant M(i=1,2)\right\} .$$

[You may use without proof any version of the Baire Category Theorem.] Deduce that, for some $K>0$, every $x \in X$ can be written as $x=x_{1}-x_{2}$ with $x_{i} \in C$ and $\left\|x_{i}\right\| \leqslant K\|x\|(i=1,2) .$